Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

Abstract

Abstract We consider the generalized Jacobian J ~ {\widetilde{J}} of the modular curve X 0 ⁢ ( N ) {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the ℚ {\mathbb{Q}} -rational torsion points on J ~ {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on J ~ {\widetilde{J}} and its Eisenstein property.